### ABSTRACT

### Vessels moored in deep water may require buoys to

support### part of the weight of the mooring lines.

### The effects that

size, and location of supporting buoys have on the dynamics### of Spread Mooring Sysems SMS at different water depths

are assessed by studying### the slow motion nonlinear

### dynamics of the system.

Stability analysis and bifurcation### theory are used to determine the changes in SMS dynamics

### in deep water based as

### functions of buoy parameters.

### Catastrophe sets in a two-dimensional parametric design

### space are developed from bifurcation boundaries, which

### separate

### regions

### of qualitatively

### different

dynamics.### Stability analysis defines the morphogeneses occurring

as bifurcation boundaries are crossed. The mathematical model### of the moored vessel consists of the horizontal plane

### -surge. sway and yaw -

### fifth-order, large drift, low speed

maneuvering equations._{Mooring lines made of chains are}

### modeled quasistatically as catenaries supported by buoys

### including nonlinear drag and touchdown.

Steady excitation### from current, wind and mean wave drift

### are tnodeled.

Numerical applications are limited to steady current and shOw### that buoys affect

_{both the static and dynamic loss of}

stability ### of the

### sYstem, and may even cause chaotic

response.### NOMENCLATURE

AC anchored catenary

CG center of gravity

D8 buoy diameter

FPSO Roatin Production Storage and Offload

h water depth

depth of AC

depth of segment S2A of SC depth of segment S2B of SC L

### length of vessel

## r

### IECNNI$CHE UP1fl1ERSngg

### Laboratorium voor

### SCheepshydrmhI

### rchief

Mekelweg_{2, 2628 CD Delit}

'ICL016- 1868/s . Fa 015 78183S
### EFFECT OF SIZE AND POSITION OF SUPPORTiNG BÔYS ON

### THE DYNAMICS OF

### SPREAD MOORING SYSTEMS

### Luis 0. Garza-Rios

### Michael M. Bernitsas

### Department of Naval Architecture

### and Marine Engineering

### University of Michigan, Ann Arbor,Michigan

total length of the mooring line (1wt = +

total lencth of AC

total length of SC/location of buoy along the mooring line

mass of vessel mass of buoy

number of mooring lines number of buoys

anchored catenary segment suspended catenary segment

segment of suspended catenary running from the buoy to the lowest point of SC

segment of suspended catenary running from the vessel to the lowest point of SC suspended catenary

Spread Moonng S)stem

horizontal tension components of SI (AC) horizontal tension components of S2 (SC) current speed

inertial reference frame/position of vessel CG with respect to inertial frame

position of the center of gravity of buoy with respect to the inertial frame

vessel fixed coordinate system buoy fixed coordinate system

coordinates of the anchoring point oat the sea floor

body fixed coordinates of the vessel faii'léad current angle with respect to (x,v,:) frame horizontal angle between the X-axis and the mooring line. In m8 n SI S2 S2A S2B SC SMS Uc (x.v,z)

### (x5,8.z8)

(X.Y.Z) (X3. Y8. ZB)### a

### Q

### I.

### INTRODUCTION

Station-keeping of ships and other floating production systems cart.' be achieved via mooring and dynamic positioning. In recent years. the

### design problem of

moorirg FPSOs has gained considerable attention as the water depth of oil reserves, gas and other natural 'resources has increased. Presently. station-keeping in deep waters

### (700 -

1800 meters) via FPSOs is generally achieved by dynamic positioning 15, 19]. while mooring takes place mostly in shallow and intermediate water depths (less than 700 meters). The next logical step toward achieving an efficient and less expensive station-keeping design would be to develop a hybrid system with dynamic positioning and mooring. To design sound mooring systems in deep waters, however, it is necessary to gain a qualitative understanding of their slow motion dynamics in such an environment.Several studies show that mooring systems dynamics strongly depend on certain design parameters, such as the number of mooring lines; the fairlead position, material, orientation, pretension of lines, etc. [1-4. 6. 7,

### 10, II. 13.

16, 18, 20]. A preliminary study of the nonlinear dynamics of Turret Mooring Systems (TMS) in relatively deep waters (on the order of 1200 meters) without buoy supports [7] shows that the system tends to lose its dynamic stability with increasing water depth. In deep waters, the weight

### exerted by the mooring lines on the

vessel and onto themselves may lead to high stresses on the vessel and mooring line breakage. This problem can be solved by### supporting the line weight by buoys, utilizing different

types of lines (such as chains, synthetic fiber ropes, and steel cables), or a combination of both. In this work, the dynamics of mooring systems with anchoring chains and supporting buoys are studied, and the effects of the position and size of the buoys on the system are assessed based on the design methodology for mooring systems developed at

### the University of Michigan since 1983 [2, 3, 4,

10, 18].Section 11 describes the mathematical model of the system, mooring line model, and external excitation. The state space representation of a spread rhooring system with supporting buoys, and the equations to solve for system equilibria are derived in Section III. In Section IV. stability analysis is performed and bifurcation theory used to assess the behavior of two different geometries of a line, four-buoy system. Catastrophe sets around the principal equilibrium position are constructed to discern the system

### dynamics for different water depths and buoy size and

position along the mooring lines. These sets reveal that the effects of buoy size and position on the system dynamics vary widely with system geometry.

### In Section V. the

feasibility of mooring in deeper waters with buoy supported chains is discussed.

### Ii. PROBLEM FORMULATION

The slow-motion dynamics of a Spread Mooring System (SMS) with chain mooring lines and their supporting buoys

### is modeled in this section in terms of the equations of

motion and the kinematics of the vessel and each supporting buoy, the mooring line catenary equations of the system,

### and external excitation consisting of time independent

current, wind and mean wave drift. The vessel hvdrodynamic maneuvering model is based on the large drift angle. low speed. fifth-order maneuvering equations [21. 22]. The

equations of motion of the system consist of the horizontal plane - surge. sway and yaw - slow motions for the vesel. and motions or each supporting buoy in surge. sway nd heave. First the mooring line model with one supporting

buoy is derived.

11.1.

### Mooring Line Model

The mooring lines of the system are model'ed quasistatically as a submerged catenary partially supported with buoys. To facilitate discussion, the concepts describ'ed in this paper apply to SMS with a single buoy supporting each cacenary. A complete SMS mathematical model with any number of buoys per line is derived by Garza-Rios and Bernitsas, 1999 [81.

Figure 1 shows the general geometry of a submerged twp. dimensional catenary whose weight is partially supported b a single buoy in water depth h. The inertial frame (x.z has its origin at the anchoring point on the ocean floor. The catenary is divided into two segments. Segment SI consists

### of the

portion### that spans from the

mooriIg terminal on the ocean floor to the supporting buoy, and is### labeled as AC (Anchored Catenary) [9].

Segment S2 connects the buoy to the floating body. is labeled as SC (Suspended Catenary) [17]. and is further divided into segments S2A and S2R. as shown in Figure 1. Segment S2A runs from the point of connection of the buoy to the Iowest point of the SC. and segment S2B runs from the lowest point 'of the SC to the point of attachment on the vessl. The geometric properties of the catenary shown in Figure 1are: P. the total length of the horizontal projection of AC:

. the length of the horizontal projection of the suspendd

part of AC: d. the horizontal distance of AC in the ground:

### 1,. the length of the horizontal projection of SC: '., tte

length of the horizontal projection of segment S2A:

### f,5,

the length of the horizontal projection of segment S2B; h. the vertical distance of the lower point of the buoy from th'e sea bed.. is also the depth of AC; h4. the vertical distanc from the lower point of SC to the lower point of the buoy. is also the depth of S2A: and ''B' the vertical distance from### the lower point of SC to the point of attachment to th

floating vessel, is also the depth of S2B. The following relations hold:

### ': =F4+P5

### h=hIh,A+h.B

The total length of the catenary ('wT) is the components F arid F.. which correspond to of segments SI and S2. respectively..

= 1w1 +

The external forces acting on the catenary of Figure I are

[8]: F05 is the buoy drag force due to current: drag force on segment SI of the catenary: F.

F!D

and

is the,

F:8D

are

### the drag components on segments S21 and S2B,

respectively; and F5. is .the force imposed by the floating:(1 (2) (31

### sum of its

the lengths. (4)vessel on the

_{moorinc line at the}

point of attachment .tôthe
vessel (an end-point condition).

_{In addition. L!.is the}

current speed

### acting in the - .r

direction.### The tension in each of the catenary

end points is given by### r.=.j72.+12.

(5)

### where T. is the tension of segment S. and 1, and 7.

are### the horizontal and vertical tension components of segthent

S.. respectively. The

### horizontal tension components of

### segments SI and S2 are therefore denoted by 7

_{arid 7,.}

### The horizontal tension components of segments S2A and

S28 are equal in magnitude. The vertical tension component### of each of the three segments is.given by [8]:

112.

_{Fquations of. Motion}

The catenary model presented in Subsection 11.1 is used

### here to model the line forces exerted on the floating vessel.

### Figure 2 shows the general geometry of a n-line SMS

showing 2 + B reference frames, where n8 is the number of buoys in the system and 11_{is the number of mooring lines}

### (B =

### n).

In Figure 2, (.t.v. ) is the inertial reference frame### with its origin located at mooring terminal

1; (X,Y,Z) is the vessel-fixed reference frame with its origin located at the### center of gravity of the vessel (CG); (X.Y'.Z) is the

### buoy-fixed reference frame of the i-ih (1

= 1,.., n5) buoy.### with X

### pointing horizontally

in the### direction of its

### anchoring point (x,v.:), and Z

pointing upwards.### Figure 2 also shows that the deformation of the two

### seements of each line, AC and SC. is two dimensional. Due

### to the drag forces acting on the buoys, however, the two

segments of each line are in different vertical planes making the overall deformation of each line three dimensional.### The slow motion dynamics of the system can be modeled

### by three equations for the horizontal plane slow motions of

### the vessel in surge. sway and yaw, and three equations of

### motion in surge, sway and heave for each buoy in the

system.### The horizontal plane equations of motion of the vessel in

surge. sway and yaw are given by### (m+m)u(m+ni)rt

=XH +### {lcos/3w

_{}

-+ 1C'surge _{(9)}= +

### j{7sini3'i

### -

### F}

(6) =### 7,.

### sinh[.2LJ.

_{(7)}=7

### Siflh[.B)

_{(8)}+ Fcwa) (lOp

### (L

### + J. )r

=_{NH +}

_{X{iY sin/3m}

_{-}

### F.}

### -

.### (II)

1=1where in

### is the mass of the vessel; ,n and m are the added

### mass terms in surge and sway; L is the second moment of

### inertia about the Z-axis;

.1... is the added moment of inertia: u. v,### and ii, v are the relative velocities and accelerations

### in surge and sway with respect to water;

### r

### and r are the

relative angular velocity and acceleration in yaw with respect to water.### For each mooring line in equations (7) - (9). (xe.

### 'ce) are the vessel-fixed coordinates of the mooring line

fairlead;_{$}

is the. horizontal angle between the X -axis and
the mooring line measured counterclockwise from the vessel:
'BDx and F.80 are the .drag forces on segment 2B of SC in
### the X and Y directions, respectively [8, 9, IS].

### In equations (9)-ill).. X.

### H' N

### are the

velocity### dependent horizontal plane hull hydrodynaniic forces and

### moment in surge. sway and yaw expressed in terms of the

large drift, slow motion derivatives. [10, 21. 22) as:XH = X,, +Xuu+LXuuu2

_{+X.vr}

_{(12)}

= }.v + +

### }.v5

+ Yurur+_{Y,.urIr1}+

_{1cjrlITI}(13) Nfl

### .'sv + Nuv +

+ +Nrr### + Ndd

### +N1,juvIr+ N.v2r

(14)In addition. Fsurge. and are the external forces

### and moment acting on the vessel due to steady wind and

### second order mean wave drift

[3].### These forces are not

### considered in the numerical applications in this study since

the focus here is the effect of buoys on SMS design.The vessel kinematics are given by

### .i= ucosi' -

### +1]. cosa ,

(15)### V=usinW+vcosW+USina ,

(16)(17)

### where (J is the current speed; a is the current angle with

### direction as shOwn in Figure 2. and

### '

### is the drift angle of

### the vessel.

The equations of motion for each of the supporting buoys. in surge. sway and heave are given by [8]:

(m

### +.4i" =

### 7j +Tjcos$'

### F'F.,

. (18)### (m'

+### )V = Tj'

### cos$

### -

### -

### -

### F,1

. (19)p,(i) (m + .4

### )(I)

### = F" - 7;

sinh(__y5_J### (piJ '

(I)" f)

### for 1=1.

### For

each buoy in the system, m9 is the buoy mass; A11,A,, and A33 are the buoy added mass components in surge,

sway and heave; u8, v8 and w8 are the buoy relative accelerations in

### surge. sway and

heave: F8 is the netbuoyancy

### force of

the buoy, i.e. the difference between itsbuoyancy and its weight; F0, and FDBY are the drag forces

on thebuoy in surge and sway [8J; F4, and LI are the drag forces on AC in the directions parallel and perpendicular to

the mooring line motion [8]; F,A and F.ADy are the drag

forces on segment S2A Of SC in the X3 and Y9 directions, respectively [8]; and $8 is the horizontal angle between the two segments of the catenary, measured coUnterclockwise

### from X8.

The 3 n8 associated

### kinematic relations of the buoy

system are [8J:

### i=ucosyvsiny+U!_cosa

(21) (I) (iI### ' =usiny+vcosy+U!L_sina

(22)### .(i)_

(I) (23) '-8### for i=l...n8.

In equations

### (21) - (23),

UB. VB and w8 are the relative velocities### of

the buoy in surge, sway and heave; YB is the### horizontal angle between the x-axis and Sl, measured

counterclockwise from the buoy: and z8 is the vertical ordinate of the center of gravity of the buoy. Notice that kinematics (21) - (23) assume a linear current profile.

### III. STATE SPACE REPRESENTATION AND

### EQUILIBRIA OF SMS

The nonlinear mathematical model for the buoy supported SMS in deep water presented in the previous section consists, of the following 6(1 + n8) equations: three equations of motion for the vessel in surge, sway and yaw (9)-(1 1); 3 n8 buoy equations of motion in surge. sway and heave (18)-(20): and their respective 3+3 n8 kinematic relations (15)-(17) and (21)-(23).

111.1.

### State Space Representation

The equations of motion and kinematic relations of the

system can be recast as a set of first order nonlinear coupled differential equations of the form

by selecting the following state space variables:

(20)

_{x =[u.}

_{. r ..v. r. ty,}

### v,

_{x'. r'. .LJI}

(m

### +A)

-(I)

### .=ucosy_vsiiyUL_cosa

### =usiny +vcosy +Ucsina

_(iJ (i)_8

### for i=l...

All . variables in equations (26) - (37) are direct or indirect functions of the state space. vector field '25).

111.2.

### Equilibria of SMS

### The equilibria of the nonlinear model of SMS are

### stationary flows of vector

field . X. and correspond tosingular points of equation (24). Equilibria can be found by setting the time derivative of the state variable vector to

zero, i.e.

(27

(28.)

i=l ...n8,.

### (2)

Then, the nonlinear model for the SMS with supporting buoys takes the form:

XH + (rn + ,n,,)rv

### {i

### cos$'

_{F,. } + F;ure}1 =

### (m+m)

(26 - (m + m )ru +### {i

sinp(s)### - F,,, } +

---### (m+m)

n NH +### x{isin$')

_{-}

_{12BD1}}- i=!

### --(l. +J)

cos$_{cb }}

(I__ +J..)
x = ucosyi - sini +U cosa
### =usinW+vcosy/+Usina

### yF=r.

7 + 7 cos$' F -(m### +A)

T4i) - (t j) (1) (i) COS -R - DR) - LI .4DY B### -

### (rn+A)

F' -, TJ

### sintI__]

Pe°### 7sinh[_L'

pç(i)

(j

### i=f(x).

(24) e is an equilibrium ofwhere the overbar on the state variable vector i denotes an

equilibrium state.

The number of equations that must be solved

simultaneously to find an equilibrium constitute the zero solution

### to equations (26).37).

The so!ution to such equations, however, requires that the tension components 1 and### i,,

be known. Therefore. additional auxiliary equations are needed to solve for the tension components required in the equations of motion and the geometrical properties of the catenaries [8. Due to state space equations (29) and (30). the equilibrium values for the vessel velocity components u and### v can be

recast### in terms of the

equilibrium drift angle t,v as f011ows:### u=ULcos(i7a)

. (39)### 7=Usin(i7a)

. (40)Furthermore, state space equations (31) and (37) reveal that the values for the vessel rotational velocity r and the buoy

### heave velocities w, (i=l,

equilibrium; i.e.

NH

### +x

### {jsinW

### _TU}

### ye,

_{v{T}

cos/3" _{-}

_{+}caw = 0

where it is implied that the functions containing i can be replaced by relatioris (39) - (41). The buoy equilibrium equations become:

(i)

_{I}

7(1
')
### -

(1r)1### [.L.J

### iflh_[

### :i)

### n8), are zero

at (45)### for i=1 ...n8.

Thus. 3 +1 1 ri _{equations must be solved for equilibrium.} _{In}

expressions (51) and (52). it has been assumed that the

### vertical position of the buoy center of gravity

z isapproximately equal to the depth of its corresponding AC (h'). The relation between these is given by

R + _{(57)}

where is the radius of the i-th buoy.

### In equations (55) and (56). for mooring line i. (.x. v)

are the horizontal plane coordinates of the anchoring point

### on the sea floor: and (i!,

.t).)_{are the horizontal plane}attachment coordinates on the vessel with respect tO the inertial reference frame (x. v).

d

### IV. STABILITY AND BIFURCATIONS OF

### EQUILIBRIA

Equilibria

_{of nonlinear dynamical}

_{systems and}

_{the}behavior of such systems near equilibrium provide local information on dynamical flows (trajectories) [12]. Such information can be useful in surmising the global beiàvior

### of a system

in_{the time domain or}

_{in}

_{a phase space.}Consequently. knowing all equilibria and - their nature provides global information ol the qualitative behavior of the system and renders time simulations

_{unnecessary}

_{in}general [1, 3. 4. 7. 18]. Further, bifurcations of equilibria

### for i=I...

The auxiliary equations (43)### sinh(..!)]

1 7(1) '\ sinhi Icoshi### -(

n9. equations (50) are### I

7(i)'\### _____,JO(zW+2__)_o

that must [8): 1 7(i### I+coshl

.4-t.27_{)}

be solved
### (

(j) Isinhi### -4k-(27j

along with (-51) 0 (52) (53) (54) (55) (56)### 27)

(i) e° _.._.i_sinhI4 P (i)### ----sinhI

U-### p

7(1) \### 27)

I \### __'

\### =o.

### (I)

### I

7(i) 2B### sinhI

### I

7(t) 2A i;:### )

P### (27)

### ((ii)2 +(1)i))2

### -7

### ii't sin ?' +

### t)cos) +U-

### sna = 0

(50Thus, the 6(1 + n8) state space equations that must be solved

simultaneously to find the SMS equilibria can be reduced to 3+5 n8- cquations. In

### addition.

_{6B auxiliary}

tension/geometry equations need tà be solved in conjunction

### with the

remaining state space equations to### find an

equilibrium [8].

The equilibrium equations for the-vessel thus become:

+ _{{?O} cos

### -

+ urge =0 (43)and their corresponding morphogeneses make it possible to

### identify areas of qualitatively acceptable dynamics.

Th.0### eliminates the need for trial and error in design of mooring

### systems.

### Based on

this### philosophy and

numerical### applications we assess in this section the effects of buoy

size and position on the design of SMS in deep water.IV.1.

### Theoretical

### Considerations

### Nonlinear stability

analysis is### used to

discern the### dynamics of specific SMS/buoy configurations.

The### stability

### properties of the system can be obtained by

performing eigenvalue analysis around all system equilibria.### If all eigenvalues of the system have negative real parts, a

### particular equilibrium is stable, and all trajectories initiated

near that equilibrium will converge toward it asymptotically.### If at

### least one eigenvalue has a positive real

part. that### equilibrium will be unstable and a small disturbance from

equilibrium will cause the system trajectories to diverge from it t23).### Bifurcation sequences are then studied to find

### qualitative changes in the dynamic behavior of the system

### by varying one or more design variables for a specific

### mooring system configuration.

Such sequences determine### regions of qualitatively similar behavior, such as stable.

unstable. periodic, and chaotic. Graphically they are plotted### in stability charts. commonly referred to as catastrophe sets.

### In this paper, catastrophe sets are constructed as functions

### of two parameters:

### the diameter of the buoys (D3). and the

### location of the buoy along the mooring line, defined by

### t,,2.

_{All buoys have the same diameter and all are placed at}

### the same location along the corresponding line.

The water depth h### is used as parameter in catastrophe sets

forh =700.

### 750 and 800 m.

### The ranges of buoy size and its

### position along the mooring line depend on a number of

factors such as the water depth and the net buoyancy force of### each buoy.

### Thus, for a particular chain mooring line in a

### specific water depth. only a certain range of combination of

### buoy size and position can be used effectively to support the

### weight of the chain.

### In this paper, rather than providing a

### complete study of size-location combination for buoys. we

### determine bifurcation sequences for specific ranges of those

parameters.### This allows us to assess the effects of the

supporting buoys on the dynamics of SMS.IV..2.

### Analysis of the SMS

### To illustrate the possible effects of buoy size and position

### on the dynamics of the system, we consider a 4-line tanker

### SMS whose geometric properties are shown in Table

1 [10)### in a 2 knot head current, a

180'.### The dimension of the

### state space is 6(1 + n5).

### For a 4-line SMS with one

### supporting buoy per line.

### n8 = n =4, and the state space

### becomes 30-dimensional.

In### this example, all moonng

### lines have the same length

### ',. = 1500 meters. with an

average breaking strength of 5159 kN [14).### In addition. to

### different geometries of the SMS (denoted as UI and G2 are

### considered, and their characteristics shown in Table 2.

The### tanker is moored in a symmetric pattern with its (X.Y)

a.'us### parallel to the (x.v) reference frame as shown in Figure 3.

### The orientation of each line in Table 2 is defined by £7. the

### horizontal angle between the vessel-fixed X -axis and the

mooring line measured counterclockwise.Table I. Properties of the tanker SMS Property

LOA (length overall) LWL (length of waterline)

### B (beam)

### D (draft)

C8 (block coefficient) 1 (displacement) m m### I--

J.-272.8 m 259.4 m 43.1 rn 16.15 m### 0.83

### I.5374x105 tons

### 9.110x106 kgs

### 1.360x

108 kgs### 7.180x 1011 kg.m2

### 5.430xl0H kg.m2

### Table 2 and Figure 3 show that the only difference

between geometries GI and G2 is in the aperture between th two forward mooring lines, which is bigger in geometry G2. Also note that the horizontal pretension component for both### segments of each catenary are equal.

### These are based ot

pretension values for the pair D8 =7 m.### f,

=700 m.### Figure 4 shows a series of catastrophe sets

for th& principal### equilibrium position of the system for

both### geometries GI and G2 in the parametric design space (D5

w2) with the following ranges

### 5mD8 7m.

### 600 m S F,,, 5 750 m .

IfOr water depths of 700. 750 and 800 meters.

Table 2. Characteristics of SMS geometries GI and G2 Characteristic Geometry GI Geometry G2 Fàirlead Coordinates (m)

### x. v

42), 2)

### The catastrophe sets of Figure 4 present three regions of

### qualitatively different dynamics, and are labeled as I, hand

Ill.

### The three lines on the right correspond to bifurcation

£7(l) _{350.00}

_{315.00}

10.00 ### 45.00

(3)_{170.00}

_{170.00}£714) 190.00 190.00 Horizontal Pretension (kN) Line I 260.5628 260.5628 Line 2 260.5628 260.5628 Line 3 104.2251 104. 225 1 Line 4 104.225 I 104.2251

### 125.00. -4.1023

### 125.0. -4.1023

### 125.00, -4.1023

### 125.0. -4.1023

### -129.69. 0.00

### - 129.69, 0.00

### -129.69. 0.00

### -129.69, 0.00

Orientation of lines (deg)boundaries for geometry GI: the remaininc two lines on the

### left correspond to bifurcation boundaries of geometry G2.

Thus, regions Gl-1 and GI-Il pertain exclusively to geometry### GI. while regions G2-1 and G2-lIl pertain to geometry G2 in

Figure 4. These regions are described as follows:Region I: The principal equilibrium of the system is stable; all 30 eigenvalues have negative real parts. A small random

### disturbance

from### the

### principal

### equilibrium

### initiates

### trajectories converging to it in forward time.

### Region II:

### The principal equilibrium of the system is

### unstable with a one-dimensional unstable manifold. i.e.

it### has a real eigenvalue with positive real part.

The system### undergoes static loss of stability when crossing from region

### I to region II. with the genesis of two additional stable

### equilibria.

### These may be stable

### or unstable for the

### parametric ranges shown in Figure 4 depending on the size

### and position of the buoy.

Stability and bifurcation analyses### around these equilibria is, however, out of the scope of this

### paper.

### Region lii:

### The principal equilibrium of the system is

### unstable with a four-dimensional unstable manifold.

Two### complex pairs of eigenvalues have positive real parts.

This### corresponds to dynamic loss of stability of the two forward

### buoys when crossing from region I to region Ill.

Since there are no other nearby equilibria to attract the trajectories. the forward buoys oscillate. Since the motions of the buoys### and the vessel are coupled. the system itself exhibits chaotic

### motions due to the

### large amplitude oscillations of the

### forward buoys.

### In simulations this

### is manifested in the

### chaotic motion of the vessel.

### In region Ill, the dimension

### of the unstable manifold is four.

### The dynamics exhibited by each geometry in Figure 4 is

detailed below:### Geometry GI:

### As shown in Figure 3, geometry GI is

characterized by the small angle of aperture between the two### forward mooring lines (20').

### The sets of three lines at the

### right of Figure 4 correspond to static bifurcation boundaries

### of geometry GI for water depths of 700, 750 and 800

meters.

### As previously mentioned, the pnnëipal equilibrium

### of the system loses its static stability when crossing from

### region I (stable) to region II (unstable), and two alternate

### stable equilibria result from such bifurcation.

### Note that for

### the ranges of parameters shown, stable region

I increases### with increasing buoy position

_{( '. buoy size (DB) and}

### water depth (h).

### Geometry G2

### This geometry is characterized by a, large

### angle of aperture of the forward mooring lines (90') as

### shown in Figure 3. and thus

it is### expected to exhibit

qualitatively different dynamics with respect to geometry GI.### The sets of two lines to the left of Figure 4 correspond to

dynamic bifurcations of geometry G2 for water depths of 750### and 800 meters.

Specifically, these lines denote boundaries### of dynamic loss of stability of the forward buoys in the

system, and occur when crossing from region I to region Ill.### The principal equilibrium of the system is thus stable to the

### right of its corresponding bifurcation line and unstable

### (chaotic) to the left.

### Since there are no other equilibria to

### attract

the### trajectories,

the### system undergoes chaotic

### (unpredictable) motions resulting ultimately

in line breaking.### Notice from Figure 4 that such unstable region

### increases as the position and size of the buoy decrease, and

### as the water depth increases.

_{As shown in Figure 4, the}

system

### is'stable for all

( DB. Ps,,) values for a water depth of 700 m.### A preliminary conclusion of the dynamics of the

### buoy-supported SMS is that geometry GI renders a more stable

### configuration, since the unstable region leads to alternate

### stable equilibria, while geometry G2 may result in chaotic

### behavior. The selected ranges of parameters (D5. t,.,. h).

### however, lead to wide variations in the depths of each

### mooring line segment (h1 for segment SI, h2A for segment

### S2A, and h8 for segment S2B), and therefore variations in

the mooring line tension. Thus, it may be impractical to try### to design a mooring system based on all parameter ranges

### for the pair (D9, ',) in Figure 4.

### Table 3 shows the

principal equilibrium values of the various depths h1 . h.,4

### and h

### for both geometries Gl and G2 at the four end

### points of Figure 4, for a water depth of 750 m. Due to the

### symmetric nature of the system, the equilibrium values of

### these variables are equal for Mooring Lines (ML) I and 2:

the same holds true for Mooring Lines (ML) 3 and 4.Table 3. Equilibrium values for vertical distances of each catenarv segment (rn). h =750 m

### One design goal. for a buoy is that the depth of neither

### catenarv segment becomes equal to the water depth. U that

### were to be the case. there would be no need to implement

buoys in the system. As shown in Table 3. neither calenary### segments depth reaches the water depth

at### any one

Geometry GI Geometry G2

### \ILI and 2 ML 3 and 4

### ML I and 2 ML 3 and 4

### (DB, ',) (m)

182.279### = (5.0,600.0)

184.771 188.033 184.754 h.1### 0.179

1.149 3.120### 0.940

h,B 567.901### 566.378

565.070 566.186 (DB.### w2) (m) = (5.0,750.0)

h1 .121.391 124.119_{118.622}

_{125.247}h 19.530 27.101 9.679 30.007 648.2 19

### 652.982

641.057 654.760### (P8. ') (m) = (7.0,600.0;

### 496.815

### 504.623

485.993 511.548 h### 94.207

102.619 72.440 115.619 h### 347.392

### 347.997

336.447 354.070 (D5, h1 2) (m)### 436.595

### = (7.0.750.0)

### 443183

424.823 451.128 '2A### 120.096

128.419 95.357 144.126 h,3_{433:5b1}

### 435.237

420.534 442.998parametric end point in Figure 4. Notice from Table 3 that with increasing distance the weight that each buoy must support increases as well. For a particular buoy size. this leads to a decrease in the '.ertical distance of the AC

### (h1) and an increase in the depths of S2A and S2B.

Conversely, as the size of the- buoy D9 increases for a particular value of P, the net buoyancy of the buoy also

### increases, thus providing an added upward force and

increasing the depth of AC. This also increases the depth of S2A (hzA) and decreases the depth of S2B( h,5). Table 3 also shows that at the two end points of Figure 4 for DB=5

m, the depths of segments S2B are excessively high. The same holds true for end point (DB

### i7 m.

e,,, =600 m) forthe values of h1. - The end point (D8=7 m. _{',,2}=750 m)
results in well distributed values for the various depths at
equilibrium, and thus a design based on these values would
be more practical. Also note from Table 3 that the lower

### left corner end point

### ( D =5 m,

es.. =600 m) results in small values for the depth h,A. In simulatiOns, the large### motions of the vessel will cause the system to lose the

geometry of Figure I. In such case, the nature of the suspended catenary is lost, resulting in excessive mooring line tensions.

Other combinations of buoy size/position in Figure 4 may result

### in inadequate mooring line

tensions, which are### proportional to the depth of the catenary.

The most practical values for DR and_{,}therefOre lie within some specific ranges of the parameters considered in Figure 4. Table 4 shows the equilibrium values of h1 . h and

### h5

forboth geometries G I

### and 02 for the pair

(D8=6.85 rn.1w2 =695 m) at a water depth of 750 m. Notice thaL the
equilibrium values for the depths of the segments are less
than those in Table 3 for the pair(DB=7 m, _{'w2} =750 m).

### This results in lower vertical mooring line tensions and

therefore are more practical. In addition, this parametric configuration is stable for both geometries, as shown in Figure 4 Other combinations of the parameters

_{(D8,}

_{2)}may result in more practical geometric configurations as well. The stability and bifurcation analysis presented in this section. however, should be used to determine whether such geometries result in sound stable systems.

Table 4. Equilibrium values for vertical distances (m). D5=6.85 m. 695 m, h 750 m

Geometry 01 Geometry 02

The results obtained in Tables 3 and 4 indicate that the practical values for the pair (D8, f,) exclude, for the most part, all ranges in the lower and left sides of Figure 4. For the remaining ranges, geometry 02 shows stable behavior of

its principal equilibrium, while geometry G I may fall in

static loss of stability, resulting in a non-svmmetric equilibrium. The design of a buoy supported SMS based n either -geometry (or any other system conliguration and catastrophe sets would require further analysis based on simulations to assure that the maximum limits of the vessel motions and the mooring line tensions are satisfied.

### V. CONCLUDING REMARKS

-A mathematical model has been developed for modeli9g

### the slow motion horizontal plane dynamics of moorirg

systems with supporting buoys: The design methodology developed at the University of Michigan fOr spread mooridg systems [1-4 10. 18] has been extended to include buoy.

This methodology makes it possible to assess the effects of buoys on the system behavior without trial and error dr repealed simulations. It is

### based on deriving

tim1eindependent properties of mooring systems with buoys such as equilibria and their stabilities, bifurcation sequences an$i their singularities, and morphogeneses across bifurcation boundaries. Numerical applications of a 4-line FPSO with one buoy per mooring line have been used to illustrate this methodology and assess the importance of buoys in mooring dynamics. It has been shown that variations in position and size of supporting buoys may lead to static and dynamic loss of stability and even to chaos, depending on the geometry of the. system. The effect of water depth on the systerii dynamics has also been studied, and it was shown that the dynamically unstable region increases with increasing wate depth. while the opposite effect is observed for staticalh unstable systems.

### ACKNOWLEDGMENTS

This work is sponsored by the

### University

o:Michigan/Industry Consortium in Offshore Engineering. anc Department of Naval Architecture and Marine Engineering

### University of Michigan.

Industry participants includt Amoco. Inc.; Conoco. Inc.: Chevron. Inc.: Exxor Production Research; Mobil Research and Development Shell Companies Foundation; and Petrobrás, Brazil.### REFERENCES

### [I]

### Bernitsas MM. and GarzaRios, L.O.,

"Effect of Mooring Line Arrangement on the Dynamics of Spreadl Mooring Systems.' Journal of Offshore Mechanics and Arctic Engineering, Vol. 118, No. 1, February 1996. pp. 7-20.### Brnitsas. M.M

and Garza-Rios. L.O.. 'Mooring System Design Based on Analytical Expressions of Catastrophes of Slow-Motion Dynamics.' Journal of Offshore Mechanics and Arctic Engineering. Vol. 119. No. 2, May 1997. pp. 86-95.### Bernitsas MM., and Papbulias.

F.A.. "Nonlinear Stability and Maneuvering Simulation of Single Point Mooring Systems,' Proceedings- of Offshore Station### Keeping Symposium. SNAME. Houston. Texas.

February 1-2. 1990. pp. 1-19.

Chung. J.S.. and Bernitsas, M.M.. "Dynamics of Two-Line Ship Towing/Mooring Systems: Bi furcations.

### ML I and 2 ML 3 and 4

### ML I and 2 ML 3 and4

h1 426.884 433.297 41&327 440.113

h,A 101.481 109.741 79. 222 123.2 18 h!B 424.597 426.444 412.895 433. 106

Equilibrium of Turret Systems.; Hydrodynamic Model and Experiments." Journal of Applied Ocean Research. Vol.. 20, 1998. pp. 145-156.

### (l4( Nippon Kaiji Kyokai . Guide to Mooring Systems.

N.K.K.. Tousei. TOkyo. 1983.

Nishimoto, K.. Kaster. F., and Aranha, J.A.P. "Full Scale Decay Test of a Moored Tanker, ALAGOAS-DICAS System." Report to Petrobrás. Brazil, February

1996.

Nishimoto, K., Brinati, H.L. and Fucatu, C.H..

### "Dynamics of Moored Tankers SPM and Turret."

Proceedings of the Seventh International Offshore and Polar Engineering Conference (ISOPE), Vol. 1. Honolulu, 1997. pp. 370-378.

[171 Papoulis. F.A.. and Bernitsas. MM.. "MOORLINE; A Program for Static Analysis of MOORing LINEs. Report to the

### University

of Michigan/Sea Grant/Industry Consortium in Offshore Engineering. and Department of Naval Architecture and Marine Engineering, The University of Michigan. Ann Arbor. Publication No. 309, May 1988.Papoulias, F.A. and Bernitsas, M.M., "Autonomous Oscillations, Bifurcations. and Chaotic Response of

Moored Vessels", Journal of Ship Research. Vol. 32. No. 3. September 1988. pp. 220-228.

Pinkster. J.A. and Nienhus. V.. "Dynamic Positioning of Large Tankers at Sea " Proceedings of 18th Annual Offshore Technology Conference. .OTC Paper No. 5208. Houston. May 1986. pp. 459-476.

120) Sharma. S.D.. Jiang. T., and Schellin. T.E.. "Dynamic

### Instability and Chaotic Motions of a

Single Point### MOored Tanker." Proceedings of the

17th ONR Symposium on Naval Hydrodynamics, The Hague. Holland, August 1988. pp. 543-562,Takashina, J..

### "Ship Maneuvering Motion due to

Tugboats and its Mathematical Model." Journal of the. Socierv of Naval Architects of Japan. Vol. 160. 1986. pp. 93-104.

Tanaka. S., "On the Hydrodynamic Forces Acting on a

### Ship at Large Drift Angles." Journal of the West

Society of Nai,al Architects of Japan. Vol. 91. 1995. pp. 81-94.

(231 Wiggins. S. Jntroduction. to Applied Nonlinear Dynamical Systems and Chaos Springer-Verlag. New York. Inc.. 1990.

Singularities of Stability Boundaries. Chaos," Jo..u:nal of S/up Research, Vol. 36. No. 2, June 1992 pp.93-105.

[5] Davison. N.J.. Thomas, N.T.. Nienhus. V. and Pinkster. J.A., 'Application of an Alternative Concept in Dynamic Positioning to a Tanker Floating Production System" Proceedings of the 19th Annual Offshore Technology Conference, OTC Paper No. 5444, Houston. April 1987, pp. 207-223.

(6] Fernandes, A.C. and Aratanha, M.. "Classical Assessment to the Single Point Mooring and Turret Dynamics Stability Problems," Proceedings of the 15th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), Vol. I-A, Florence, Italy, June 1996, pp. 423-430.

Garza-Rios, L.O., and Bernitsas, M. M., "Nonlinear
Slow Motion Dynamics of Turret Mooring Systems in
Deep Water." Froceedin.gs of the 8th International
Conference on the Behaviour of Offshore Structures
(BOSS), Delft, the Netherlands, July 1997, _{pp.}

177-188.

GarzaRios, L.O. and Bernitsas, M.M., "Slow Motion Dynamics of Mooring Systems in Deep Water With

### Buoy Supported Catenary Lines." Report # 339.

University

### of Michiganl Industry

ConsOrtium in### Offshore

Engineering, Department### of

Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor. January 1999.[91 Garza-Rios. L.O., Bernitsas, M.M. and Nishimoto. K.,

"Catenary Mooring Lines With Nonlinear Drag and Touchdown," Report # 333, University of Michigan. Department

### of Naval

Architecture and Marine Engineering. Ann Arbor, January 1997.[tO] Garza-Rios, L.O.. Bernitsas, M.M., Nishimoto. K. and Masetti.. I.Q., "Preliminary Design of Differentiated Compliance Anchoring Systems," Journal of Offshore Mechanics and Arctic Engineering, Vol. 121, No. I, February 1999. pp. 9-15.

[II] Gottlieb. 0.. and Kim, S.C.S., "Nonlinear Oscillations,

### Bifurcations and Chaos in a Multi-Point Mooring

### System with a Geometric Nonlinearity," Journal of

Applied Ocean Research. Vol. 14. 1992. pp. 832-839. [12] Guckenheimer. J.. and Holmes. P..

### Nonlinear

Oscillations. Dynamical Systems, and .Bifurcations of Vector Fields. Springer-Verlag, New York, Inc., 1983. [131 Leite, .A.J.P.. Aranha, J.A.P., Umeda, C. and de Conti,

Figure 1. Geometry of buoy-supported catenary

### GEOMETRY Gi

Figure 3. Geometric characteristics of Ci and G2

hgure -4. _{Catastrophe set of buoy supported SMS} _{- Geometries Gi and G2}

750 725-700-'. 675 -650